Mathematics Question on Quadratic Equations

Question

If x = log (y +√y2 + 1 ) then y =

Answer 1

The correct option is: (C) sinh x.

y =sinh(x), let's start by solving the equation x =log(y +y 2+1) for y.

Given equation: x =log(y +y 2+1)

First, let's eliminate the logarithm by exponentiating both sides with base 10: 10 x =y +y 2+1

Now, isolate the radical term on one side: 10 x − y =y 2+1

Square both sides of the equation to eliminate the square root: (10 x − y)2=y 2+1

Expand the left-hand side: 102 x −2 xy +y 2=y 2+1

Now, simplify the equation: 102 x −2 xy =1

Isolate y on one side: 2 xy =102 x −1 y =2 x 102 x −1

Now, we'll express sinh(x) in terms of exponential functions: 2sinh(x)=2 ex − e − x

Comparing this expression with y =2 x 102 x −1, we notice a similarity if we substitute ex for 10 x.

Therefore, we have: y =2 xe 2 x −1

Since y =2 xe 2 x −1 and sinh(x)=2 ex − e − x are equivalent expressions, we can conclude that y =sinh(x).

Hence, the answer is justified: y =sinh(x).